Optimal Extraction of Geothermal Resources

摘要

In this paper we propose a mathematical model for optimizing the way energy is extracted from a geothermal reservoir over time. The model was formulated as a profit maximization problem, with linkages to resource-related functions such as well decline and recharge. A single-tank reservoir model coupled with a wellbore simulator was used to obtain the well decline curves. Several scenarios were investigated, to determine the effects of placing restrictions on the production rate and the lease time for the resource. Analytical solutions were found for simplified versions of the problem while the more complicated scenarios were solved using numerical methods. This model indicated that the optimal behavior pattern would be to extract large amounts of energy in the early stages (first 10 to 15 years) of utilization of the resource, drilling occasional makeup wells to maintain maximum power output for the power plant. As more energy was extracted from the resource, the wells would decline in productivity, to the point where it became uneconomical to drill more make-up wells. After that time, the production from the resource would be governed entirely by the well decline curves, which in turn depended on the state of the resource. Eventually the combined output from the wells became equal to the recharge (which also depended on the state of the resource), at which point sustainable (steady state) extraction was reached. Our investigation indicated that the optimal extraction pattern for the short-term owner (say a 30-year lease holder) and a long-term lease owner (say a government wanting to exploit the resource in a sustainable manner), would be very similar, although the short-term owner would stop drilling make-up wells at a slightly earlier time. The effects of reducing the maximum production capacity to something relatively close to the maximum sustainable extraction rate seemed to lead to a marginally smaller sustainability level, and considerably smaller profits. These results of our modeling were particularly interesting when compared to the extraction process that has taken place in The Geysers geothermal field. They indicate that perhaps the management of The Geysers was not disadvantageous as some might believe, but instead, that market forces may have pushed the development towards an economical optimum.