Generalization of the multirate basis for time convolution to unequal forward and reverse rates and connection to reactions with memory

Abstract

[1] The convolution form used to express mass transfer between mobile and immobile aqueous domains, and often associated with multirate mass transfer representations of matrix diffusion processes, is generalized to the case of unequal forward and reverse rates for each of the multiple rates of the multirate mass transfer and is shown to represent also linear but non-Markovian reactions that kinetically partition mass between mobile and immobile phases with rate of return to mobile phase dependent on contiguous time spent in the immobile phase. In the case where a multirate model refers to multiple sites with distributed mobilization or release rates but single-valued immobilization rate, an equivalent formulation is found using single-site mobilization-immobilization with nonMarkovian mobilization rate dependent on contiguous time spent immobilized.Citation: Ginn, T. R. (2009), Generalization of the multirate basis for time convolution to unequal forward and reverse rates and connection to reactions with memory, Water Resour. Res., 45, W12419,

Upscaling colloid transport and retention under unfavorable conditions: Linking mass transfer to pore and grain topology

“…[], as shown in Dentz and Berkowitz [], who point out that both models can be cast in the same integrodifferential equation form. Furthermore, Ginn [] showed a modified form of the MRMT can be cast in the form of the exposure time model of Ginn []. In summary, all three of these approaches are different ways of accommodating a multiplicity of reentrainment rates of colloids.…”

“…In fact, this factored form of the CTRW is mathematically equivalent to the well-known multirate mass transfer modeling (MRMT) approach of Haggerty et al [2000], as shown in Dentz and Berkowitz [2003], who point out that both models can be cast in the same integrodifferential equation form. Furthermore, Ginn [2009] showed a modified form of the MRMT can be cast in the form of the exposure time model of Ginn [2000aGinn [ , 2000b. In summary, all three of these approaches are different ways of accommodating a multiplicity of reentrainment rates of colloids.…”

Pore-Scale Process Coupling and Effective Surface Reaction Rates in Heterogeneous Subsurface Materials

“…Slow diffusion process allows intra-granular solutions to evolve close to mineral equilibrium that can deviate from that in bulk solutions. Geochemical reactions such as sorption/desorption (Ball and Roberts 1991;Qafoku et al 2005;Liu et al 2008;Ginn 2009) and mineral precipitation and dissolution (McGreavy et al 1992;Lee et al 1998;Anderson et al 2002;Liu et al 2006;McKinley et al 2006McKinley et al , 2007Pallud et al 2010a) can occur in defi ance of thermodynamic conditions in bulk solutions. The slow diffusion, on the other hand, decreases mass exchange rate with external pore waters, leading to non-ideal behavior of chemical transport even for non-reactive species (Beven and Germann 1982;Ball and Roberts 1991;Ewing et al 2010Ewing et al , 2012Hay et al 2011).…”

Exposure-time based modeling of nonlinear reactive transport in porous media subject to physical and geochemical heterogeneity

“…By introducing the exposure time as additional dimension, and considering that a sample is a weighted average over different exposure times, the author tried to explain inconsistencies of advective-dispersivereactive transport in the presence significant chemical heterogeneity. While the author has applied the approach as "generalized exposure-time" concept to various problems of transport (e.g., Ginn, 2009), it comes with the key disadvantage of increasing the computational effort from 3-D or 4-D spacetime to a 4-D or 5-D domain (see also the Laplace-transfrom technique to evaluate the exposure-time distribution developed by Cornaton, 2012). In the present study, we thus do not consider full exposure-time distributions at a single point in space and time, and restrict the analysis to the mean exposure time, which can be evaluated by a slightly modified version of the mean groundwater-age equation (Goode, 1996).…”