5:15 PM - 6:45 PM
[SCG52-P01] Solving subseaflor hydrothermal circulation using the explicit finite element method and its applications to flow within seamounts: solving slow flows fast Part 3.
Keywords:permeable flow, seamount, numerical method
The modelling of low-temperature hydrothermal circulation in the submarine crust requires numerical calculations of thermal convection using the Darcy law. The equation is similar to the incompressible flow of the Navier-Stokes (NS) equation (the viscous term in the NS equation is replaced by a coefficient proportional to the velocity). The solution method is also similar, using an implicit time integration method. Another important feature is that the subsurface has a complex structure. For example, seamounts, which are typical sites where the hydrothermal circulation occurs, protrude impermeable sediment to expose the flat seafloor. Treating such a system with complex geometry using the ordinary finite difference method suffers from severe limitations (a finite difference method would treat the slope of the seamount surface as a staircase shape), and it is desirable to use the finite element method with a flexible geometry. In this presentation, the finite element method is considered for solving thermal convection in seepage flows. Here, an explicit method rather than an implicit method is used as the time integration method, and the Runge-Kutta method is also used to speed up the integration.
Several improvements are needed to apply the explicit method to the finite element method. Although the equation finally obtained in the finite element method formulation is similar to that obtained by the difference method, there are two main differences. The first one is that the coefficients on each dependent variable are non-uniform, reflecting geometric irregularities. The equation cannot be solved efficiently by the explicit method as it is. Rather, it would be constrained by the time step determined by the finest grid width. Second, the time derivative term includes the surrounding grid points (in the difference method, this is equivalent to approximating the zero-th order derivative of a physical quantity at a point by a weighted average of the surrounding points). In this case, although the time evolution is explicit, a simultaneous linear equation should be solved. To relax these constraints, the following procedures are implemented. For the first point, a factor depending on the geometry of the finite element is applied to the time derivative term, since geometrically inhomogeneous systems have a similar structure to systems with inhomogeneous viscosity coefficients, etc. For the second point, a mass centralisation is performed. In other words, the weighted average of the surrounding points for the zero-th order derivative is replaced by the pointwise information at the point of interest.
We perform two stages of calculations with different levels of computational complexity to examine the effectiveness and limitations of the explicit method. The first stage involves hydrothermal circulation in an inclined permeable layer sandwiched between impermeable layers. The calculations are performed in the two-dimensional domain using quadrilateral elements. The regular grid is continuously deformed, and the elements are irregular in shape but regular in sequence. In this case, if the coefficients on the time derivative term are selected appropriately, the finite element method can be solved in a manner comparable to the application of the explicit method to the finite difference method. At present, the method for selecting the coefficients is trial and error and needs to be devised on the basis of stability analysis. The second step considers the hydrothermal circulation through a seamount protruding from the impermeable sediment. Calculations are carried out in three-dimensional space using tetrahedral elements. The elements are created using the freeware "Tetgen". This calculation can be solved efficiently if the elements are created properly. On the other hand, if extremely small elements are created, the calculation speed is severely limited, although the calculation itself can be performed as it is an explicit method. When using the explicit method to calculate the hydrothermal circulation of a system with complex geometry, such as a flow involving a seamount, using the explicit method, the creation of the descretisation is of particular importance. This is a different property from the implicit solution method, which may work when it is solved.
Several improvements are needed to apply the explicit method to the finite element method. Although the equation finally obtained in the finite element method formulation is similar to that obtained by the difference method, there are two main differences. The first one is that the coefficients on each dependent variable are non-uniform, reflecting geometric irregularities. The equation cannot be solved efficiently by the explicit method as it is. Rather, it would be constrained by the time step determined by the finest grid width. Second, the time derivative term includes the surrounding grid points (in the difference method, this is equivalent to approximating the zero-th order derivative of a physical quantity at a point by a weighted average of the surrounding points). In this case, although the time evolution is explicit, a simultaneous linear equation should be solved. To relax these constraints, the following procedures are implemented. For the first point, a factor depending on the geometry of the finite element is applied to the time derivative term, since geometrically inhomogeneous systems have a similar structure to systems with inhomogeneous viscosity coefficients, etc. For the second point, a mass centralisation is performed. In other words, the weighted average of the surrounding points for the zero-th order derivative is replaced by the pointwise information at the point of interest.
We perform two stages of calculations with different levels of computational complexity to examine the effectiveness and limitations of the explicit method. The first stage involves hydrothermal circulation in an inclined permeable layer sandwiched between impermeable layers. The calculations are performed in the two-dimensional domain using quadrilateral elements. The regular grid is continuously deformed, and the elements are irregular in shape but regular in sequence. In this case, if the coefficients on the time derivative term are selected appropriately, the finite element method can be solved in a manner comparable to the application of the explicit method to the finite difference method. At present, the method for selecting the coefficients is trial and error and needs to be devised on the basis of stability analysis. The second step considers the hydrothermal circulation through a seamount protruding from the impermeable sediment. Calculations are carried out in three-dimensional space using tetrahedral elements. The elements are created using the freeware "Tetgen". This calculation can be solved efficiently if the elements are created properly. On the other hand, if extremely small elements are created, the calculation speed is severely limited, although the calculation itself can be performed as it is an explicit method. When using the explicit method to calculate the hydrothermal circulation of a system with complex geometry, such as a flow involving a seamount, using the explicit method, the creation of the descretisation is of particular importance. This is a different property from the implicit solution method, which may work when it is solved.