We have developed a numerical code, called SCHAFF, which can treat problems involving slightly compressible fluid and heat transfer in multidimensional porous media. Solutions to the appropriate partial differential equations are obtained by the integrated finite difference method which is essentially equivalent to making mass and energy balances over finite subregions or elements. The resultant system of finite difference equacions is solved by an iterative procedure and solutions to the fluid flow and energy equations are coupled by interlacing in time so that the temperature and velocity fields are interdependent. The useful concepts of fluid and thermal time constants as indicators of nodal response times and numerical stability limits are an inherent part of the numerical scheme.

In applying the numerical model to the problem of circulatory convection in saturated porous media, we have discussed the relevant aspects as they pertain to geothermal systems and show in Fig. 1 that results from SCHAFF on the relationship between the Rayleigh number and the dimensionless heat transfer coefficient or Nusselt number are in good agreement with numerical and experimental results from other authors. We then used the numerical model to extend these results to include the effects of temperature dependent parameters and density variations with pressure. Variations in fluid viscosity and thermal expansivity with temperature result in substantial differences in the values of the critical Rayleigh number for the onset of convection and the Rayleigh number-Nusselt number relationship compared with corresponding constant parameters results (Fig. 2 ). However, consideration of fluid density as a function of pressure produced no noticeable effect on convective motion.

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Copyright 1975, Stanford Geothermal Program: Readers who download papers from this site should honor the copyright of the original authors and may not copy or distribute the work further without the permission of the original publisher.