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

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

Valentina Prigiobbe^{1},
Marc A. Hesse^{2,3}, Steven L. Bryant^{1,3}

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

^{2}Department of Geosciences,
University of Texas at Austin, USA.

^{3}Institute 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 (Sr^{2+}) as a function of the longitudinal coordinate. Sr^{2+}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 Sr^{2+}, 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 Sr^{2+}. 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 Sr^{2+}. 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 Sr^{2+}
arises ahead of the retarded Sr^{2+} front which travels at the average
fluid velocity. These conditions define the necessary conditions for the
occurrence of the non-classical reactive transport of Sr^{2+}, 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 Sr^{2+} retarded front predicted by the theory
and an isolated pulse of Sr^{2+} traveling at the average fluid
velocity (Figure 2).** **

**Figure 2**.
Measured
concentration of Sr^{2+} (a, b) and of Na^{+} (b) towards the
pore volume injected (PV). It is possible to see that the pulse of Sr^{2+ }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., Ba^{2+},
Ca^{2+}, Mg^{2+}, Co^{2+}, and Ni^{2+}, which
are characterized by an adsorption isotherm similar to Sr^{2+}, 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.

**References **

[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.

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