Abstract
The basic equations governing the behavior of a two-phase mixture of carbon dioxide (CO2) and water are discussed. A Newton-Raphson scheme, based on the alternating direction implicit (ADI) method for multidimensional problems, is used to solve the nonlinear finite difference approximation of the governing nonlinear system of partial differential equations. Sample calculations showing the behavior of hypothetical reservoirs with varying CO2 contents are presented.
Introduction
Geothermal reservoirs often contain an appreciable amount of noncondensable gases that have major effects on the behavior of the reservoir in both its natural state and under exploitation. In its natural state, the partial pressure of the noncondensable gas causes the reservoir to boil at a lower temperature than does a pure water field. Under exploitation the presence of CO2 or hydrogen sulfide (the most presence of CO2 or hydrogen sulfide (the most common gases in geothermal fields) dominates the transport and thermodynamical characteristics of the flow. It already has been shown that for lumped parameter models of geothermal reservoirs, small parameter models of geothermal reservoirs, small differences in the CO2 content of the reservoir cause major changes in the pressure, enthalpy, and gas content of the discharge fluid. Because of the importance of the gas content in influencing the design of flash turbines and other geothermal energy conversion systems, it is essential to include the effects of noncondensable gases in simulations of gassy geothermal reservoirs.Most of the procedures developed for simulating the behavior of geothermal reservoirs have features that are not modified easily to include the effects of a noncondensable gas. These difficulties arise either from the numerical procedures used or from the methods of treatment of the thermodynamics of the system. The work of Pritchett et al. uses the fluid (mixture) density and internal energy as primary dependent variables and interpolation between tabular values for determining thermodynamic properties. With the addition of a noncondensable properties. With the addition of a noncondensable gas, the use of density and internal energy (together with one other unknown) leads to an indirect iterative calculation of the other fluid properties such as temperature and pressure. Even when the noncondensable gas is not present, the use of pressure as one of the unknowns appears to be desirable.The procedure developed by Faust and Mercer uses the same unknowns used here, namely pressure and mixture enthalpy, but their procedure for solving the finite difference equations involved requires some modification to allow the introduction of the extra unknown arising from the presence of a noncondensable gas. The numerical methods used by Mercer and Faust, Thomas, and Coats all are based on the earlier work of Price and Coats. Basically the problem involves implicit nonlinear finite difference approximations of the governing partial differential equations that are solved using the partial differential equations that are solved using the Newton-Raphson method. For multidimensional problems the matrices that arise in the problems the matrices that arise in the Newton-Raphson procedure are sparse but have a large bandwidth. Careful ordering of the equations enables the bandwidth to be reduced significantly. Still greater numerical efficiency is possible by using the alternating direction procedure combined with the lagging of corrections to the permeability terms in the Newton-Raphson process, The application of this ADI procedure to petroleum reservoir problems is restricted severely by petroleum reservoir problems is restricted severely by stability limits on the time step, but compressibility and thermal expansion effects in geothermal problems tend to stabilize the scheme. problems tend to stabilize the scheme. JPT
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Mean square convergent three points finite difference scheme for random partial differential equations
