Transport with spatially variable kinetic sorption: recursion formulation A. K. Mishra 1 a , *, A. Gutjahr b & H. Rajaram c a Department of Earth and Environmental Science, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA b Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA c Department of Civil, Environmental and Architectural Engineering, University of Colorado, Boulder, CO, USA (Received 19 May 1997; revised 2 April 1998; accepted 2 June 1998) A recursion formulation for the transport of linearly sorbing solutes undergoing non- equilibrium sorption is developed. Constant or spatially varying sorption kinetics can be modeled using the recursion approach. The sorption and desorption rates are modeled as two independent random processes with a prescribed mean and covariance structure with spatial variability in the rate parameters included as well. The recursion solution, in terms of the probability density function for solute travel times, is derived by specifying transition probabilities for moving between the aqueous and sorbed phases. A few simple examples are used to illustrate the approach. The computer implementa- tion leads to a very rapid algorithm that is easily extended to cover cases beyond the basic model presented here.  1999 Elsevier Science Ltd. All rights reserved. Keywords: recursion formulation, sorption kinetics, spatial variability, solute trans- port, heterogeneity of sorption and desorption rates, Markov process. 1 NOMENCLATURE C(y) Covariance function at a lag y F Mean of ln k f process K d Distribution coefficient k f Sorption rate coefficient, [day ¹1 ] k r Desorption rate coefficient, [day ¹1 ] L Length of the column [m] P j n, k Probability that a solute particle takes n space steps to move k time steps starting from state j; j ¼ 1 aqueous state; j ¼ 2 sorbed state r 11 Probability of remaining in the aqueous phase r 22 Probability of remaining in the sorbed phase r 12 Transition probability to move from the aqueous to the sorbed phase r 21 Transition probability to move from the sorbed to the aqueous phase v fluid velocity [m/day] Dt Time step [day] Dx Space step [m] m k f Mean of k f [day ¹1 ] j 2 Variance of ln k f or ln k r l Correlation length [m] 2 INTRODUCTION Several contaminants found in the subsurface exhibit a tendency to adsorb onto the aquifer solids. The adsorption of organic and metallic species is vividly observed in recent large-scale field tracer tests such as the Borden 21 and Cape Cod tracer tests. 10,25 The influence of adsorption on con- taminant transport is often modeled by using a distribution coefficient (K d ), which strictly applies only in case of linear equilibrium adsorption. However, significant evidence has accumulated over the last decade suggesting that adsorption isotherms for several common contaminants are non- linear, 2,29 and furthermore, that the rates of adsorption are relatively slow. 11,16,18,19 The transport of sorptive con- taminants in subsurface environments is further complicated by heterogeneity in physical and chemical properties of natural earth materials. For instance, Robin et al. 22 estimated a variance of 0.52 for ln K d , and significant spatial correlation in ln K d variations at the Borden site. Several recent theoretical and computational studies have focused on reactive transport in heterogeneous media. Valocchi 28 presented an analytical solution for reversible linear kinetic sorption in a stratified aquifer. Cvetkovic and Shapiro, 6 Selroos and Cvetkovic, 23 and Dagan and Cvetkovic 9 Advances in Water Resources Vol. 22, No. 5, pp. 549–555, 1999  1999 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0309-1708/99/$ - see front matter PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 2 3 - 2 549 *Corresponding author. E-mail: akmishra@lbl.gov 1 Present address: Earth Sciences Division, E.O. Lawrence Berkeley National Laboratory, Berkeley, California, USA.