We introduce a solver method for spatially dependent and nonlinear fluid transport. The motivation is from transport processes in porous media (e.g., waste disposal and chemical deposition processes). We analyze the coupled transport-reaction equation with mobile and immobile areas.
The main idea is to apply transformation methods to spatial and nonlinear terms to obtain linear or nonlinear ordinary differential equations. Such differential equations can be simply solved with Laplace transformation methods or nonlinear solver methods. The nonlinear methods are based on characteristic methods and can be generalized numerically to higher-order TVD methods [Harten A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 1983, 49(3), 357–393].
In this article we will focus on the derivation of some analytical solutions for spatially dependent and nonlinear problems which can be embedded into finite volume methods. The main contribution is to embed one-dimensional analytical solutions into multi-dimensional finite volume methods with the construction idea of mass transport [Geiser J., Mobile and immobile fluid transport: coupling framework, Internat. J. Numer. Methods Fluids, 2010, 65(8), 877–922]. At the end of the article we present some results of numerical experiments for different benchmark problems.
If the inline PDF is not rendering correctly, you can download the PDF file here.
 Abramowitz M., Stegun I.A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1992
 Bear J., Dynamics of Fluids in Porous Media, Enviromental Science Series, American Elsevier, New York, 1972
 Bear J., Bachmat Y., Introduction to Modeling of Transport Phenomena in Porous Media, Theory Appl. Transp. Porous Media, 4, Kluwer, Dordrecht, 1991
 Davies B., Integral Transforms and their Applications, Appl. Math. Sci., 25, Springer, New York-Heidelberg, 1978
 Eykholt G.R., Analytical solution for networks of irreversible first-order reactions, Water Research, 1999, 33(3), 814–826 http://dx.doi.org/10.1016/S0043-1354(98)00273-5
 Frolkovič P., Geiser J., Discretization methods with discrete minimum and maximum property for convection dominated transport in porous media, In: Numerical Methods and Applications, Borovets, August 20–24, 2002, Lecture Notes in Comput. Sci., 2542, Springer, Berlin, 2003, 445–453
 Geiser J., Discretisation Methods for Systems of Convective-Diffusive Dispersive-Reactive Equations and Applications, PhD thesis, Universität Heidelberg, 2004
 Geiser J., Discretization methods with embedded analytical solutions for convection-diffusion dispersion-reaction equations and applications, J. Engrg. Math., 2007, 57(1), 79–98 http://dx.doi.org/10.1007/s10665-006-9057-y
 Geiser J., Mobile and immobile fluid transport: coupling framework, Internat. J. Numer. Methods Fluids, 2010, 65(8), 877–922 http://dx.doi.org/10.1002/fld.2225
 Geiser J., Zacher T., Time dependent fluid transport: analytical framework. preprint available at http://webdoc.sub.gwdg.de/ebook/serien/e/preprint_HUB/P-11-05.pdf
 Higashi K., Pigford T.H., Analytical models for migration of radionuclides in geologic sorbing media, Journal of Nuclear Science and Technology, 1980, 17(9), 700–709 http://dx.doi.org/10.3327/jnst.17.700
 Kelley C.T., Iterative Methods for Linear and Nonlinear Equations, Frontiers Appl. Math., 16, SIAM, Philadelphia, 1995 http://dx.doi.org/10.1137/1.9781611970944
 LeVeque R.J., Finite Volume Methods for Hyperbolic Problems, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, 2002 http://dx.doi.org/10.1017/CBO9780511791253
 Van Genuchten M.T, Convective-dispersive transport of solutes involved in sequential first-order decay reactions, Computers & Geosciences, 1985, 11(2), 129–147 http://dx.doi.org/10.1016/0098-3004(85)90003-2