| Title | Pressure-enthalpy formulation for numerical simulations of supercritical water/steam systems applied to a reservoir in Tuscany, Italy |
|---|---|
| Authors | Büsing, H; Niederau, J; Clauser, C |
| Year | 2016 |
| Conference | European Geothermal Congress |
| Keywords | pressure-enthalpy-formulation, numerical simulation, two-phase water/steam system |
| Abstract | We implement a water pressure-enthalpy formulation within the high-level system Firedrake for the automated, portable solution of partial differential equations using the finite element method (FEM). Our formulation consists of a mass balance equation for the two phases water and steam. This equation is complemented by a single energy balance equation, assuming local thermodynamic equilibrium. Using Firedrake allows us to employ a broad variety of space discretizations for arbitrary partial differential equations (PDEs). Applying the Unified Form Language (UFL) and FEniCS Form Compiler (FFC) Firedrake accomplishes the parallel FEM assembly based on PyOP2. To this end, we choose a lowest-order discontinuous Galerkin method for the space discretization on hexahedral grids which is equivalent to a two-point flux approximation finite volume scheme. This enables an effective use of the high-level abstractions of Firedrake for interfacing with a finite volume solver. For the time discretization we use the implicit Euler method. The system of nonlinear PDEs of the pressure-enthalpy formulation is strongly coupled. But the unconditionally stable implicit Euler method still allows us to take large time steps. Our nonlinear algebraic systems are solved with Newton’s method. The derivatives of the Jacobian are automatically calculated via automatic differentiation (AD), a transformation which yields exact derivatives up to machine precision. Finally, the linear systems are solved with PETSc, providing a front-end to many different solvers and preconditioners. Next to the standard choice of BiCGStab with incomplete LU preconditioning, we can also use advanced solvers like algebraic multigrid methods (AMG). Then, we apply our formulation to a supercritical water/steam reservoir located in Tuscany, Italy. We will show numerical simulations of this reservoir including phase changes from water to steam and conditions ranging from liquid/gas to supercritical ones. This research takes place within the EU project DESCRAMBLE (Drilling in dEep, Super-CRitical AMBients of continentaL Europe). |