On vertical diffusion of gases in a horizontal reservoir

Exact and approximate solutions to vertical diffusion in gravity-stable, ideal gas mixtures in gas reservoirs, depleted oil reservoirs, or drained aquifers are presented, and characteristic times of diffusion are obtained. Our 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 the reservoir, with no flow boundaries at the top and bottom, is filled with CO₂ and the upper part with CH₄. 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 integrated mass balance method. 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. Furthermore, we consider diffusion of CO₂ with partitioning into connate water. We show that at reservoir pressures the CO₂ retardation by water cannot be neglected. The diffusion-retardation problem is modelled by a non-linear diffusion equation whose self-similar solution is obtained. Finally, we obtain a self- similar solution to a nonlinear diffusion problem. This solution is a good approximation at early times, before the diffusing gases reach considerable concentrations at the top and bottom boundaries of the reservoir.