A two-dimensional (areal) finite-difference geothermal model is presented, and results are compared with a Galerkin, finite-element model previously developed by the authors. In both previously developed by the authors. In both models, it is assumed that local thermal equilibrium exists and capillary pressure is negligible. The continuity equations for steam and water are reduced to two non-linear, partial differential equations posed in terms of fluid partial differential equations posed in terms of fluid pressure and enthalpy. These equations are pressure and enthalpy. These equations are applicable to both hot-water (single-phase) and vapor-dominated (two-phase) geothermal reservoirs
In both the finite-difference and finite-element models the time derivatives are approximated by finite-difference expressions. For each time step, this yields a system of nonlinear equations. The nonlinear coefficients are treated as constants during each time step and are evaluated using either the values of pressure and enthalpy at the beginning of the time step or values extrapolated to the mid-time level. In the finite-difference model the solution is improved by Newton-Raphson iteration. This technique markedly reduces mass and energy balance errors while permitting larger time steps. The linearized equations in both models are solved by a Gauss-Dolittle procedure for nonsymmetric, banded matrices. Several example problems, based on analytical solutions and published experimental data, are used to evaluate the relative merits of finite-difference and finite element techniques applied to geothermal reservoir simulation. Results suggest that the finite element method is more suitable for hot-water reservoirs due to less numerical diffusion and better approximations of boundary and internal geometries (with fewer node points). For vapor dominated systems, the finite-difference model appears superior because it reduces mass and energy balance errors and exhibits less numerical oscillation.