The goal of this paper is to find exact and approximate solutions
to vertical diffusion in gravity-stable, ideal gas mixtures in gas
reservoirs, depleted oil reservoirs, or drained aquifers. Using
these solutions, we have obtained estimates of the characteristic
times of diffusion. These solutions also can be used to test
numerical simulators that model diffusion after gas injection.

First, we consider isothermal, countercurrent vertical diffusion
of carbon dioxide and methane in a horizontally homogeneous
reservoir.  Initially, the bottom part of a uniform, homogeneous
rock with no flow boundaries at the top and bottom is filled with
CO2 and the upper part with CH4.  Both gases are treated as
ideal. At time equal zero, the two gases begin to diffuse. We
obtain the exact solution to the initial and boundary-value
problem using Fourier series method.  For the same problem, we
also obtain an approximate solution using the integral method
proposed earlier by the fourth author. The latter solution has a
particularly simple structure, provides a good approximation and
retains the important features of the exact solution. Its
simplicity allows one to perform calculations that are difficult
and non-transparent with the Fourier series method. It also can be
used to test numerical algorithms.

Second, we consider diffusion of CO2, with partitioning into
connate water.  We show that that at reservoir pressures, the
CO2 retardation by water cannot be neglected.  The
diffusion-retardation problem is reduced to a non-linear diffusion
equation whose self-similar solution is obtained.

Third, we obtain a self-similar solution to a nonlinear diffusion
problem.  This solution provides a good approximation to the exact
one before the diffusing gases reach considerable concentrations
at the top and bottom boundaries of the reservoir.