| Abstract |
Simulation of heat production in high-enthalpy geothermal systems is associated with a complex physical process when the cold water is invading the steam-saturated control volumes. Because of phase transition and the large variation in thermodynamic properties between liquid water and steam, the nonlinear numerical solution of governing equations in fully-implicit approximation can experience difficulties commonly known as negative compressibility. In the process of solution, due to the steam condensation, the reduction in fluid volume reduces the pressure in the control volume. It creates a region in parameter space where the gradient of the residual equation does not point to the direction of the solution. If initial guess of the nonlinear problem is located in this region, the solution based on Newton's update cannot converge and needs to be repeated for a smaller timestep. This problem brings simulation to the stalling behaviour where nonlinear solver wastes a lot of computations and performs at very small timestep. To tackle this problem, we formulate continuous localization of Newton's method based on Operator-Based Linearization (OBL) technique. In OBL approach, the continuous operators in the governing equations, related to different physical mechanisms (e.g. convection or conduction), are translated into multi-dimensional tables. In the course of simulation, only the supporting points evaluated based on the reference physics when points between them are interpolated. This provides a unique mechanism where nonlinear physics can be represented at different scales of accuracy. The coarser is the representation, the more linear operators become. In our continuous localization approach, we start with the coarse approximation of the governing physics in pressure-enthalpy parameter-space. Due to the more linear form of operators, only a few nonlinear iterations reduce residual below the predefined tolerance. Next, the OBL approximation is modified towards the reference nonlinear physics and a few more iterations bring the residual below the tolerance again. With the refinement in physics, the solution will gradually approach the final solution where residual will satisfy the convergence criteria of the reference physics. This continuous localization in physics avoids the negative compressibility phenomena since the problem at the coarser approximation has a unique gradient pointing towards the correct solution and helps to localize the solution for higher resolutions in the region of monotone gradient behaviour. As a result, simulation can perform at larger timesteps in comparison to the traditional nonlinear solution. |