Material balance calculations often referred to as â??zero-dimensionalâ?¿ reservoir evaluations have been used extensively in petroleum and geothermal engineering. This paper presents a Havlena and Odeh-type1 material balance depletion model for two-phase reservoirs incorporating adsorption phenomena. A straight line, formed between groups of thermodynamic and adsorption properties of water and the cumulative production history provides the initial-fluid-in-place and the size of the vapor-dominated â??steam capâ?¿. Adsorption phenomena were found by Economides and Miller2 as the controlling mechanism in the depletion of vapor-dominated geothermal reservoirs. A material balance for vapor-dominated geothermal reservoirs, demonstrating the importance of adsorption phenomena is also presented. A straight line provides the initial-fluid-in-place.
The general approach to the material balance depletion model was first suggested by Schilthuis3. The equation provides a volumetric balance between the expansion of reservoir fluids as a result of the cumulative production.
Havlena and Odeh1 using the Schilthuis approach developed linear expressions of the material balance equation for a variety of cases including undersaturated, gas cap drive and solution gas drive reservoirs. A similar development for the two-phase geothermal reservoir still be presented in this paper.
Studies of reservoir and production behavior of vapor-dominated geothermal systems have focused on estimates of resource size. Whiting and Ramey4 presented an application of material and energy balances to geothermal steam production. Ramey5 applying conventional techniques for natural gas reservoirs, attempted to estimate the reserves of steam-in-place.
Economides and Miller2 using the experimental results of Hsieh6 in which the importance and magnitude of adsorption was demonstrated, introduced a new approach for material balance calculation of vapor-dominated systems. As it will be shown here desorption is the controlling mechanism in the depletion of vapor-dominated geothermal systems and the major variable in their material balance calculations.
The Form of the Material Balance Equation for a Two-Phase Geothermal Reservoir
As in the Havlena and Odeh1 approach a volume balance in reservoir cubic feet (m3) may be written (in their work they used reservoir barrels):
Underground withdrawal (ft3, m3)
Expansion of liquid water and boiled off water (ft3, m3)
Expansion of steam cap (ft3, m3)
Expansion of desorbed water (ft3, m3)
Reduction in the pore volume (ft3, m3)
The reduction in the pore volume will be neglected in the case of two-phase and vapor-dominated systems as has been demonstrated by Havlena and Odeh1 for saturated reservoirs. Figure 1 is a schematic depiction of the two-phase geothermal model used in this analysis.