Methods of Solution of the Equations for Convection in Porous Media, with Geothermal Applications

Abstract

Various approaches to the solution of the equations of thermal convection in fluids may be classified,, for convenience, under such headings as: amplitude instabilities, for which R/Rc : 1, where R is the Rayleigh number and Rc i sits critical value for neutral stability ; (2) the Galerkin method, a well-known numerical technique utilizing truncated expansions in orthogonal functions, which has been applied up to R/Rc z O(10); variational method, which seeks to establish bounds on the heat transport, for given Rayleigh number, up to large R/Rc; of the convection equations, usually infinite-difference form, up to R/Rc 2 O(10). ( 1 ) the Stuart-Watson method, which deals with the behavior of finite- (3) the (4) direct numerical solution

These techniques are considered in relation to the equations of convection of variable-viscosity fluid in a porous medium.

Since the particular application is intended to be geothermal convection, many simplifications may have to be accepted. First, it is assumed that the flow can be treated as flow through porous media. This is not necessarily true, although the approximation becomes more satisfactory if only large- scale motions are considered. Secondly, the medium may not be isotropic. This is not a serious difficulty, but isotropy will be assumed for convenience. Thirdly, salt may be transported as well as heat, and can exert an influence upon fluid buoyancy. Evidently, the transport of salt would involve a straightforward generalization of the treatment for heat transport (although some new phenomena are encountered), and is not considered here. Fourthly, chemical interaction of the fluid with the medium, which would introduce great complications, is assumed to be negligible .