One-dimensional oxygen diffusion into soil with exponential respiration: analytical and numerical solutions

A numerical model for one-dimensional diffusion of oxygen into soil which incorporates a soil respiration function that decreases exponentially with depth, is compared with an analytical solution. The results show that the assumption of zero flux at the bottom boundary for the numerical solution means that the assumed depth to the bottom boundary, zn, is a critical parameter. The analytical solution shows that this boundary condition leads to two distinct oxygen concentration profiles. The first occurs when the oxygen concentration approaches a constant value, C∗, as z → ∞. The second occurs when the oxygen concentration goes to zero at some depth, L. With the analytical solution, the nature of the profile can be determined a priori and either C∗ or L calculated. With the numerical solution it is not possible to determine which of these profiles will occur a priori. This can only be obtained by increasing zn and noting what happens to the oxygen concentration profile. sults illustrate the care needed in using numerical solutions and the extra physical insight that may be gained from using analytical solutions where they are available.