Abstract
A linear closed-form analytical solution for the radial flow of steam toward a producing well in a vapor-dominated geothermal reservoir is compared with a finite difference solution to the nonlinear equations of fluid flow and energy change. Assumptions used in the development of the equations are that (1) the liquid phase, initially uniformly distributed within the reservoir, is immobile, (2) the relative permeability to steam is constant (3) local thermal equilibrium exists within the reservoir, (4) temperature changes are due only to phase change, and (5) effects of vapor-pressure lowering are negligible. With the onset of production, vigorous vaporization of liquid water in the reservoir near the wellbore creates a dry region that increases in volume as production continues. This behavior produces a circular moving boundary that separates superheated steam in the dry zone from saturated steam in the wet zone. The rate of movement of this boundary, the pressure drawdown, and the temperature and saturation distributions are obtained analytically by applying the solution to the linearized equations of flow in radial coordinates. Results obtained numerically using a finite difference solution to the nonlinear equations of fluid flow and energy agree closely with the analytical approach in spite of the nonlinearities involved.
Introduction
Methods of analysis of transient pressure data from geothermal steam reservoirs until recently have been limited to techniques developed for noncondensible gas reservoirs (for example, see Ramey). These techniques generally assume that isothermal conditions prevail in the reservoir. In vapor-dominated geothermal reservoirs, steam and liquid water are believed to coexist, with the liquid component being relatively immobile (White et al.). Under conditions of fluctuating pressure, phase changes result, and the assumption of isothermal flow is no longer valid. Therefore several authors have included nonisothermal flow theory in the analysis of steam pressure transients in porous media. Evidence for liquid vaporization within the reservoir is indirect; however, there are laboratory experiments, well test data, and chemical studies that support the contention. Moench and Atkinson presented results of a numerical study of transient pressure behavior in geothermal steam reservoirs. Their study involved a finite difference solution to the nonlinear equations of radial fluid flow and energy change. They found that it was possible to explain some, but not all, of their computational results by combining the energy and flow equations, thereby deriving an equation similar to the gas flow equation but with an apparent hydraulic diffusivity that was many times reduced. This report extends the analysis to a linear closed-form analytical solution for pressure drawdown, fully accounting for the results of the numerical study.
Mathematical Model
The geothermal reservoir is conceptualized as a horizontal aquifer partially filled with liquid water in which steam flows radially toward a discharging well. With the onset of production, vigorous vaporization of liquid water in the reservoir near the wellbore creates a dry region that increases in volume as production continues.
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