Three-Dimensional Geothermal Reservoir Simulation (1978)

Abstract This paper presents the development of a three-dimensional, finite-difference model for the simulation of geothermal reservoirs. The model is designed to simulate geothermal reservoirs that contain water in any of its vapor or liquid states and provisions are included for properly treating changes provisions are included for properly treating changes of state during a time step. This logic provides for a stable calculation of state change and eliminates pressure, heat balance, and material-balance errors. pressure, heat balance, and material-balance errors. Mass and energy balances are solved simultaneously using an implicit pressure-explicit saturation (IMPES) formulation. An implicit treatment of production rates, capillary pressure, and transmissibilities is included as an option. Thus, entire field, cross-sectional, or individual well studies can be performed efficiently. Example problems are presented to demonstrate the model's utility and to provide insight into the nature of geothermal production under various conditions. In particular, an example is presented of a reservoir initially containing subcooled liquid where fluid conditions near the production well range from subcooled liquid to saturated steam and then to superheated steam. Introduction Geothermal energy represents an essentially untapped, alternate source of energy world-wide. Exploration for this type of energy has increased, however, and should lead to the discovery and development of new geothermal areas. Notable producing geothermal fields include Wairekei in producing geothermal fields include Wairekei in New Zealand, Cero Prieto in Mexico, Matsukawa in Japan, Larderello in Italy, and The Geysers in California. Recently, industry literature has dealt with zero-(one-cell), one-, and two-dimensional, geothermal reservoir simulation. Most studies treat steam in only one state or mention the difficulty of simulating blocks that change from one state to another. Whiting and Ramey presented a zero-dimensional, geothermal model that included water influx. They discussed reservoir performance during the various states that steam can assume. Their study also included a successful match of the pressure/ production performance of the Wairekei geothermal production performance of the Wairekei geothermal reservoir in New Zealand from 1956 to 1966. Coats developed a three-dimensional, finite-difference, steam-flooding model that simultaneously solved the mass and energy equations. An IMPES formulation was used to solve for pressures implicitly followed by a solution for temperatures, saturations, and so forth. Saturations calculated using explicit rates, capillary pressures, and transmissibilities were recalculated at the end of each time interval, using an implicit formulation for these variables. Brigham and Morrow discussed the P/Z behavior of geothermal steam reservoirs. Their model was similar to Whiting and Ramey's model, but they allowed for the presence of a vapor and liquid zone. Mass and energy balances were written for both zones, and logic for a constant or falling liquid level was included. Martin analyzed internal steam drive in geothermal reservoirs by assuming that temperature, pressure, and fluid saturation gradients as well as capillary pressure and gravity were negligible. Gould described the development of a vertical two-phase, pressure-drop calculation for flow in geothermal wells. He discussed the effect of heat transfer to the surrounding formation and the effect of steam quality on production. Mercer and Faust developed a two-dimensional (areal), two-phase geothermal model using a Galerkin finite-element formulation in space and finite-difference formulation for time. They chose pressure and enthalpy as dependent variables and pressure and enthalpy as dependent variables and solved for these two variables simultaneously. They presented a 10-month simulation of a hypothetical hot-water reservoir with initial conditions similar to those in the Wairekei reservoir. No provisions were included during the course of a run for changing states - for example, from subcooled liquid to saturated steam. SPEJ P. 151