274578 Non-Classical Reactive Transport of Solute Through a Porous Medium

Tuesday, October 30, 2012: 4:05 PM
413 (Convention Center )
Valentina Prigiobbe, Petroleum and Geosystems Engineering, University of Texas at Austin, Austin, TX, Marc Hesse, Jackson school of geosciences, University of Texas at Austin, Austin, TX and Steven Bryant, Petroleum and geosystems engineering, University of Texas at Austin, Austin, TX

Non-classical reactive transport of a solute through a porous medium

Valentina Prigiobbe1, Marc A. Hesse2,3, Steven L. Bryant1,3

1Department of Petroleum and Geosystems Engineering, University of Texas at Austin, USA.

2Department of Geosciences, University of Texas at Austin, USA.

3Institute for Computational Engineering and Sciences

Field observations [1] have shown that the transport of radionuclides may be much faster than predicted by standard models applied for the description of transport of dilute solutes in reactive porous media which are based on the theory of chromatography [2]. Numerical simulations motivated by these field observations showed that under certain conditions radionuclides can be strongly retarded as predicted by the theory but also travel without retardation (Figure 1). This transport without retardation is a notable exception from the theory and represent a non-classical reactive transport behavior [3,4,5] which is complement to the well-known fast transport pathways such as colloid-facilitated transport and flow in fractures.

Figure 1. Concentration of strontium (Sr2+) as a function of the longitudinal coordinate. Sr2+front:  (a) The numerical simulation consists of a shock, a rarefaction, and the non-classical wave or pulse. (b) The analytical solution from the theory of chromatography consists of a spreading wave followed by a shock wave.

In this work, we analyze the non-classical reactive transport behavior using 1D model for incompressible flow through an iron-oxide porous medium of Sr2+, H+, Na+, and Cl-. We combine the theory of hyperbolic systems of conservation laws [6] with surface complexation and we derive the mathematical framework for the analysis of the system and the definition of the necessary conditions under which the non-classical reactive transport behavior arises [5]. Under the assumption of Na+ and Cl- conservative, local chemical equilibrium, negligible hydrodynamic dispersion, and incompressible flow, the mathematical problem reduces to a strictly hyperbolic 3x3 system of conservation laws for effective anions, which are defined as the difference of the conservative species, the total protons, given by the difference of H+ and OH-, and Sr2+. The mass conservation laws have the nonlinearity in the accumulation term due to adsorption and they are coupled by the two Langmuir isotherms of H+ and Sr2+. One characteristic field is linearly degenerate while the other two are not genuinely nonlinear due to one inflection point in the adsorption isotherms.

For this system, we solved the Riemann problem (constant initial and injected states) and the analytical solution consists of three waves separated by two intermediate points and comprises nine combinations of rarefactions, shocks, shock-rarefaction, and contact discontinuity. The slow and the intermediate waves are either a rarefaction, or a shock, or a shock–rarefaction while the fast wave is a contact discontinuity. Highly resolved numerical solutions at large Péclet numbers show excellent agreement with the analytic solutions in the hyperbolic limit (negligible hydrodynamic dispersion) except under certain conditions when a pulse of Sr2+ arises ahead of the retarded Sr2+ front which travels at the average fluid velocity. These conditions define the necessary conditions for the occurrence of the non-classical reactive transport of Sr2+, which we verify in the laboratory performing column-flood experiments [7]. In the absence of colloids and fractures, the experiments confirmed this non-classical behavior with a strongly Sr2+ retarded front predicted by the theory and an isolated pulse of Sr2+ traveling at the average fluid velocity (Figure 2).







Figure 2. Measured concentration of Sr2+ (a, b) and of Na+ (b) towards the pore volume injected (PV). It is possible to see that the pulse of Sr2+ travels at the same speed of the conservative cation Na+.

This non-classical reactive transport behavior is due to the interaction of hydrodynamic dispersion and of a strongly pH-dependent adsorption. The strong pH-dependence makes the retarded front unstable to small perturbations due to hydrodynamic dispersion which broadening the retarded front leads to the formation of the unretarded pulse.

These results raise important questions regarding the prediction of the migration of toxic compounds in the subsurface, e.g., Ba2+, Ca2+, Mg2+, Co2+, and Ni2+, which are characterized by an adsorption isotherm similar to Sr2+, and may have implications for the design of industrial chromatographic separation processes. Furthermore, it raises interesting questions for the theory of hyperbolic systems of equations, as it appears that the vanishing diffusion solution does not approach the diffusion free (hyperbolic) limit uniformly in all cases.


[1] Olsen et al. (1986) Geochim Cosmochim Acta 50, 593-607.

[2] Appelo (1996) Reviews in Mineralogy 34, 193-227.

[3] Rhee et al. (1989) First-order partial differential equations, theory and application of hyperbolic systems of quasilinear equations II Prentice-Hall.

[4] Bryant et al. (2000) Ind Eng Chem Res 39, 2682-2691.

[5] Prigiobbe, Hesse, Bryant (2012) Transport Porous Med 93, 127-145.

[6] Lax (1957) Comm Pure Appl Math 10, 537-566.

[7] Prigiobbe, Hesse, Bryant (2012) Geophys Res Lett, submitted.


Extended Abstract: File Not Uploaded
See more of this Session: Mathematical Modeling of Transport Processes
See more of this Group/Topical: Engineering Sciences and Fundamentals