Tardif d'hamonville, Pierre (2006) Modeling and simulation of advective and diffusive transport in monophasic and biphasic porous media. PhD thesis, ENPC.

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Abstract

described by advective, diffusive and dispersive fluxes. The two latter fluxes are formulated using a diffusion tensor and a dispersion tensor. The actual value of these tensors depends on the pore geometry and on the magnitude of the advection velocity.
First we are interested in the monophasic case in which the medium is saturated with an incompressible liquid. Using the double scale expansion technique, the velocity field at the pore scale is first obtained by solving the Stokes problem. Then, the velocity field is used in a vector-valued advection diffusion problem from which solution the diffusion and dispersion tensors are evaluated. We consider a finite element approximation at the microscopic scale and we perform the whole numerical analysis. For the Stokes problem, we compare three finite element spaces in terms of the divergence of the discrete velocity field. As an application, we compute the values of the diffusive and dispersive tensors for cubic and centered cubic sphere networks and we study the influence of the velocity magnitude and the pore morphology on the tensors.
This methodology is extended to the biphasic case in which we consider the vapor transport in equilibrium with liquid menisci localised in the pores. We show the influence of the vapor concentration on the menisci geometry and the importance of the pore size for the liquid-vapor equilibrium. We perform numerical simulations for cubic and centered cubic sphere networks to compute the diffusive and dispersive tensors coefficients. Finally, a maeroscopic transport problem is solved in order to understand the non linear effects due to the multi-scale nature of the problem.

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