Analytical solutions to the three-dimensional radio-nuclide transport equation for computer code verification (Q1083330)

This study is related with the crucial problem of the disposal of radioactive waste in geological formations. It is concerned with verification of the numerical accuracy of models for the transport of a single radio-nuclide in a three-dimensional permeable medium. In this way the author uses the following equation \[ (1)\quad \partial /\partial t\quad (\phi KC)+\nabla \cdot (qC)=\nabla \cdot (\phi \hat D\cdot \nabla C)-\mu \phi KC, \] where \(C=\) dimensionless nuclide concentration in pore wate, \(\phi=\) porosity, \(K=\) retardation factor, \(\hat D=\) dispersion tensor, \(\mu=\) decay rate and \(q=\) Darcy velocity. The equation (1) is solved by the method of separation of variables, i.e., \(C(x,t)=T(t)C(x)\) which results in two equations (2)\ T'\(=-\Lambda T\) \((\Lambda =\) separation constant) and (3)\ \(\nabla \cdot (uC)-\nabla \cdot (\underline D\cdot \nabla C)+\mu C=\Lambda C\) with \(u=q/K\phi\) and \(\underline{D}=\hat D^-/K.\) Then, two examples of effective dispersion tensors are considered which allow equation (3) to be separated further: a case of anisotropic diffusion (D is taken to be a constant diagonal tensor) and a case of velocity-dependent dispersion tensor associated to the following form of \(\underline{D}:\) \[ D= \begin{pmatrix} D_ x \\ &D^ y \\ &&D^ z \end{pmatrix} + \begin{pmatrix} \hat d_ x| u_ x| \\ &\hat d_ y| u_ y| \\ &&\hat d_ z| u_ z| \end{pmatrix} . \] In both cases, a series of analytical solutions to the three-dimensional equation is carefully derived and used to verify the numerical accuracy of the finite-element code NAMSOL. Good agreement is found for grid Péclét numbers \((Pe=L| q| /\phi (| \hat D|\), \(L=typical\) element length-scale) of order unity or less, but the solutions become unstable above this, as expected from standard stability criteria.