Nonlinear Convection Problems in Geology

Abstract

Many problems in geology involve the relative displacement of a fluid or liquid with the solid portion of a rock. As far as chemical exchange is concerned, there may be no need to consider two separate classes of functions (i.e. concentrations of components in the fluid and in the solid) provided there is a tendancy toward local equilibrium: both concentrations are then connected by a relation: this one is generally non linear. The simplest transport equation that may be written on such a basis is of the form c_t + f(c)_x = 0 where c and f are concentrations (scalars or vectors); this equation is very rich: it may particularly provide a framework within which such salient features as the appearance of discontinuities may be understood.

The numerical modeling allows one to simulate several aspects such as front propagation, washing out of heterogeneities, local increase of concentrations and so on. Other features such as oscillatory behaviour need to consider additive terms like diffusion terms Df(c)_xx or chemical kinetics terms. The starting problem is set in x and t variables but one is led to find the “stationary” evolution in the concentration space only thanks to a minimum entropy production principle.