  We investigate complex boundary conditions of the miscible displacement
system in two and three space dimensions with the commonly-used
Bear-Scheidegger diffusion-dispersion tensor, which describes, e.g., the porous
medium flow processes in petroleum reservoir simulation or groundwater
contaminant transport. Specifically, we incorporate the no-flux boundary
condition for the Darcy velocity to prove that the general no-flux boundary
condition for the transport equation is equivalent to the normal derivative
boundary condition of the concentration, based on which we further prove
several complex boundary conditions by the Bear-Scheidegger tensor and its
derivative. The derived boundary conditions not only provide new insights and
distinct properties of the Bear-Scheidegger diffusion-dispersion tensor, but
accommodate the coupling and the nonlinearity of the miscible displacement
system and the Bear-Scheidegger tensor in deriving the first-order optimality
condition of the corresponding optimal control problem for practical
application.
