Analytical solution and simplified analysis of coupled parent-daughter steady-state transport with multirate mass transfer

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1 WATER RESOURCES RESEARCH, VOL. 49, , doi:.9/wr8, 3 Analytical solution and simlified analysis of couled arent-daughter steady-state transort with multirate mass transfer Roy Haggerty Received 5 August revised 9 November acceted 3 November ublished 5 January 3. [] In this technical note, a steady-state analytical solution of concentrations of a arent solute reacting to a daughter solute, both of which are undergoing transort and multirate mass transfer, is resented. Although the governing equations are comlicated, the resulting solution can be exressed in simle terms. A function of the ratio of concentrations, In (daughter/arent P), can be used as a metric of mass transfer and reaction if the reactions are mostly confined to the immobile domain, where P is the ratio of roduction of daughter to decay of arent. This metric is alied with the resazurin-resorufin (Raz-Rru) tracer system in a stream to obtain an integrated measure of resiration that occurs on or in a stream bed. The sloe of the grah of In (Rru/Raz P) versus advective travel time is a function of the strength of surface bed interaction and resiration. This grah can be used for raid comarison of different exeriments and streams. Citation: Haggerty, R. (3), Analytical solution and simlified analysis of couled arent-daughter steady-state transort with multirate mass transfer, Water Resour. Res. 49, doi:.9/wr8.. Introduction [] Reactive or decaying solutes in the environment, e.g., nutrients, tracers, radionuclides, and contaminants, are commonly exchanged between mobile and immobile domains and may aroach steady-state concentrations at sufficiently long time with constant boundary conditions. In this technical note, an analytical solution for a arent comound undergoing a first-order reaction to a daughter comound is resented, which can also undergo first-order reaction. Both comounds exerience multirate mass transfer with an immobile domain [e.g., Haggerty and Gorelick, 995 Haggerty et al., Dentz and Berkowitz, 3 Luo et al., 8]. Reactions can occur in both the mobile and immobile domains however, the analytical solution may be of most use when reactions are limited to the immobile domain. The ratio of the daughter to arent concentration is a simle function that is sensitive only to advective travel time, mass transfer, and reaction. [3] Some related solutions are as follows: () steadystate transort of a single solute with first-order decay and immobile-domain diffusion [Cunningham and Mendoza- Sanchez, 6] () steady-state transort of two solutes with first-order decay [McLaren, 969 Burnell et al., ] (3) transient transort of a decay chain [Sun et al., 999] extended to diffusion into a slab [Sun and Buscheck, 3] and generalized to comounds with differing retardation factors [Srinivasan and Clement, 8]. College of Earth, Oceanic, and Atmosheric Sciences, Oregon State University, Corvallis, Oregon, USA. Corresonding author: R. Haggerty, College of Earth, Oceanic, and Atmosheric Sciences, Oregon State University, 4 CEOAS Admin Bldg., Corvallis, OR , USA. (haggertrgeo.oregonstate.edu). American Geohysical Union. All Rights Reserved /3/WR8. Model and Solution [4] A model for transient solute transort and first-order reaction along a one-dimensional flow ath where water is exchanged between mobile and a distribution of immobile zones is described by equations () (4). These equations modified from Argerich et al. [] for a metabolically active transient storage fraction of and a distribution of exchange rates as described by Haggerty et al. [] or Luo et al. [8]. The integrodifferential terms result from distributions of exchange rates. R C t Z S ðþ R t ¼ Q C A x A x a S ðþ AD C x ðþd i C () S ðþ ¼ ðc S ðþþ a t S ðþ << () C t Z S ðþ R a t S ðþ a ya S ðþ ðþd i C i yi C ¼ Q C A x A x AD C x S ðþ R ¼ ðc S ðþþ a t S ðþ a ya S ðþ << where subscrits and indicate arent and modeled daughter, C is stream concentration of the arent (mol L ), C is stream concentration of the daughter (mol L ), S () is storage domain concentration (mol L ) of the arent, S () is storage domain concentration (mol L ) of the (3) (4) 635

2 daughter, suerscrit a refers to a reaction rate associated with the storage domain (relatively active ), suerscrit i refers to a reaction rate associated with the stream (relatively inactive ), a is the rate coefficient for the decay of arent to all daughters (h ), a is the modeled daughter reaction rate coefficient (h ), y is the conversion factor between arent and modeled daughter ( ), is the robability density function of mass transfer exchange rates (h), is the ratio of storage to stream mass at equilibrium ( ), R is retardation due to sortion ( ) with subscrits indicating arent and daughter, D is the disersion coefficient (m h ), Q is discharge (m 3 h ), and A is cross-sectional area of the stream (m ). If y a or y i ¼, then arent is converted only to the modeled daughter, whereas if y a or y i <, some arent is simultaneously converted to another daughter that is not modeled. The values, S,andS are functions of however, we subsequently dro the functional notation. Here, it is assumed that the reaction only haens in the aqueous hase, a valid assumtion for biological reaction [McMurry,,. ] but not for radioactive decay. Radioactive decay would require retardation factors in the decay terms above. [5] During a constant injection to lateau concentrations, the equations eventually simlify to steady state after lateau is reached: Z C i a S d ¼ Q C A x A x AD C (5) x S ¼ a C (6) Z C i a S a ya S d i y i C (7) ¼ Q C A x A x AD C x S ¼ a C a ya a S : (8) Using a first-tye ustream boundary condition (C ðx ¼ Þ ¼ C b, C ðx ¼ Þ ¼ C b ) and concentration of at infinite distance, the after aroriate substitutions are as follows: Z " # C i a a a C ya a ac d Z 3 6 a ya a d i 7 4 yi 5C ¼ Q C A x A x AD C x () [6] Defining three effective reaction rate functions, Z f ¼ i a a d () (the effective rate of decay of C ), f ¼ i a Z (the effective rate of decay of C ), and f 3 ¼ a a ya Z a ya Z a d () a ad i yi a d (3) (the effective rate of roduction of C from C ), equation () can be rewritten as follows: f C f 3 C ¼ Q C A x A x AD C : (4) x [7] The velocity is v ¼ Q/A, the Peclet number is Pe ¼ vl/d, L is the downstream distance to the end of the reach. All arameters are assumed to be uniform, although solution could be accomlished with arameters that vary in sace using searation of variables. Results would be similar to ours excet that a number of arameters would be defined as integrals over sace and that the validity of Pe would need careful evaluation. [8] The solution for C (in equation (5)) is identical to solutions that are well known because the steady-state equation is the same as the Lalace-transformed transient equation [cf., Valocchi, 985] and because the distribution of mass transfer rates is not a function of sace: where C ¼ C b ex ½mf ð ÞŠ (5) mf ðþ¼ Pe s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4f (6) Pe ¼ x v (7) ¼ x L : (8) Observation at x is assumed, and ¼ ifx ¼ L. The solution to C is as follows: C ¼ C b ex ½mf ð ÞŠ Pe ðex ½mf ð ÞŠ ex ½mf ð ÞŠÞ C b f 3 : Pe f ½Pe mf ð ÞŠmf ð Þ (9) 636

3 [9] The ratio of C to C is given by the following equation: HAGGERTY: TECHNICAL NOTE C ¼ C b ex ½mf ð Þ mf ð ÞŠ C C b Pe f 3 fex ½mf ð Þ mf ð ÞŠ g : mf ð ÞðPe mf ð ÞÞPe f () [] The structure of () can be seen more clearly if we assume Pe : C ¼ C b f 3 ex ½ ðf f ÞŠ fex ½ ðf f ÞŠ g C C b f f or ln C f 3 ¼ ðf f Þ ln C b f 3 C f f C b f f (a) (b) 3. Alication to Resazurin [] These relationshis are now used for the resazurinresorufin (Raz-Rru) tracer system, which is sensitive to water column-bed exchange and resiration [Argerich et al., Gonzalez-Pinzon et al., ]. Raz transforms to Rru almost exclusively at the bed or in the sediment [Haggerty et al., 8], which means that i and i can be neglected. The ratio of effective rate functions simlifies tremendously and becomes a function only of reaction arameters: f 3 ¼ ya a f f a P: () a This ratio can be thought of as the rate of roduction of C relative to the rate of decay of C, which we will call the roduction-decay ratio, P. If reaction is isolated to the storage zones, P is indeendent of the mass transfer rate and relative quantity of storage zone. The effects of mass transfer are isolated within two effective decay coefficients as follows: e ¼ Z a a d and e ¼ Z a a d (3) Furthermore, for Pe= e and Pe= e (see Cho [97] and Burnell et al. [] greater than about 6 is sufficient), the equations can be simlified. Equation (b) therefore becomes: ln C P C ¼ ð e e Þ ln C b P : (4) C b [] Equation (4) is very useful. In a stream, the mass transfer effects (e.g., bed exchange, resulting in resiration) are confined to the sloe of the line, ( e e ). When used with Raz-Rru, the sloe of the line is therefore an integrated measure of bed exchange and ecosystem resiration. Although the details of bed exchange may contain many Figure. Plots of equation (4) using a known and an aroximate value of P. The known (true) value of P is.667, and the aroximated value of P is.5. The aroximation error is 33% but the ( e e ) sloe error is about 9%. The lines were generated from a full simulation of the advection-disersion-mass transfer reaction equations with arameters D ¼ 36 m h v ¼ 5 m h ¼.3 i ¼ 9:89 4 h i ¼ :75 3 h a ¼ h a ¼ :h y i ¼ y a ¼.6 and constant A. The distribution of rate coefficients was single rate, i.e., () ¼ () with ¼ 4h. Injection was constant for h and C,b /C,b ¼.5. The reaction and mass transfer arameters yield effective reaction rates e ¼.8 h and e ¼.98 h. The correct value of ( e e ) from the simulation is. h. different rates and volumes which may be nonunique, the sloe of the line can be comared from one location to another or from one time to another. [3] The roduction-decay ratio P is often known or can be aroximated with little consequence. For examle, the arent may decay only to the daughter, and the daughter may be stable, in which case P ¼. If the arent decays to other daughters at the same rate as the daughter decays, P ¼. For the Raz-Rru system, our work to date suggests that P is of order.3.8 [Haggerty et al., 8, 9 Argerich et al., and data in rearation for ublication]. If the estimate of P is incorrect, (4) is nonlinear. For this reason, large errors in P will generate a nonlinear grah and large errors will therefore be obvious. However, small errors in P can result in nearly linear grahs and therefore generate errors in the sloe of the line that are not obvious. [4] To check if the sloe of the line (4) recovers the correct value of ( e e ), the full transort equations () (4) were solved using STAMMT-L [Haggerty and Reeves, ]. Figure shows the full results, using the known value P ¼.667 (blue) and using an incorrect value P ¼.5 (red). When P was known, the sloe of the line ( e e ) was almost exactly correct. The correct value of the sloe was., and the small error was due to the aroximations of zero disersion and no reaction in the water column in the grahed function. [5] Using (4) with field data from an exeriment reorted by Argerich et al. [] gives results shown in Figure. Briefly, Raz was injected on 7 8 Setember 7 to steady-state concentration in the bedrock reach of watershed 3 at the H. J. Andrews Exerimental Forest, 637

4 Figure. Plot of equation (4) for field results given in Argerich et al. []. The three sets of markers result from using different assumed values of the roduction decay ratio P. The resulting line sloes are shown in the figure. From a full simulation of transort [see Argerich et al., ], the sloe of the line should be.5 h. Oregon. Concentrations were measured at eight locations in the bedrock reach over a distance of 56.5 m. The average velocity of water in the reach was 3.5 m h. As erfect knowledge of P did not exist, the results were lotted for three different values: P ¼.4,.5, and.6. This generates a sloe of ( e e ) ¼.95 to.34 h Argerich et al. used a temoral moments method to calculate the value of.5 h. [6] The results in Figures and suggest that the sloe of In (C /C P) indicates or at least qualifies the amount resiration and the amount of interaction between the water column and the bed. Furthermore, In (C /C P) is easy to grah and thus different sites or conditions can be comared without numerical modeling. Deending on the assumtions that can be made and the urose of the exeriment, the results could be sufficiently accurate that no other modeling is necessary. Of course, the estimate could always be imroved with more comlete use of the data and numerical modeling. [7] The results of this note are alicable to any tracer or comound during steady-state transort that is undergoing first-order decay in the aqueous hase to a second comound. An examle of a stream tracer, resazurin, was given. The results are also alicable to radionuclides, rovided that they are weakly sorbing so that most of the decay haens in the aqueous hase. The results are also alicable to contaminants in groundwater. Notation A cross-sectional area of the stream (m ). C stream concentration of the arent solute (mol L ). C,b ustream boundary condition for arent solute (mol L ). C stream concentration of the daughter solute (mol L ). C,b ustream boundary condition for daughter solute (mol L ). D disersion coefficient (m h ). f effective decay rate function for arent solute (h ). f effective decay rate function for daughter solute (h ). f 3 effective roduction rate function for daughter solute (h ). L reach length (L). m(f) function (see (6) for definition) ( ). () robability density function of mass transfer exchange rates (h). P roduction-decay ratio (see equation ()) ( ). Pe Peclet number ( ). Q discharge in the stream (m 3 s ). R retardation factor for arent solute due to sortion ( ). R retardation factor for daughter solute due to sortion ( ). S () storage domain concentration of the arent solute as a function of (mol L ). S () storage domain concentration of the daughter solute as a function of (mol L ). t time (h). v velocity of the stream (m h ). x satial coordinate (L). y a conversion factor between arent and modeled daughter in the storage zone ( ). y i conversion factor between arent and modeled daughter in the water column ( ). first-order rate coefficient for exchange between the stream and the storage zone, and equal to the inverse ofthemeanresidencetimeinthestoragezone(h ). ratio of storage to stream mass at equilibrium, and equal to As/A as defined in most transient storage literature ( ). a rate coefficient for the decay of arent to all daughters in the storage zone (h ). i rate coefficient for the decay of arent to all daughters in the water column (h ). e effective decay coefficient for the arent solute (h ). a rate coefficient for the decay of daughter solute in the storage zone (h ). i rate coefficient for the decay of daughter solute in the water column (h ). e effective decay coefficient for the daughter solute (h ). advective travel time (h). dimensionless distance defined in (8) ( ). [8] Acknowledgments. Financial suort for this work was rovided by the National Science Foundation (EAR ). Data and facilities were rovided by the H.J. Andrews Exerimental Forest research rogram funded by the National Science Foundation s Long-Term Ecological Research Program (DEB 8-338), U.S. Forest Service Pacific Northwest Research Station, Oregon State University, and the Hollis Dole Fund in Environmental Geology. The author thanks Jud Harvey, Ricardo Gonzalez-Pinzon, Olaf Cirka, and Jesus Gomez for their comments on the drafts of this manuscrit. References Argerich, A., R. Haggerty, E. Martí, F. Sabater, and J. Zarnetske (), Quantification of metabolically active transient storage (MATS) in two reaches with contrasting transient storage and ecosystem resiration, J. Geohys. Res., 6, G334, doi:.9/jg379. Cho, C. M. (97), Convective transort of ammonium with nitrification in soil, Can. J. Soil Sci., 5(3),

5 Burnell, D. K., J. W. Mercer, and L. S. Sims (), Analytical models of steady-state lumes undergoing sequential first-order degradation, Ground Water, 5, Cunningham, J. A., and I. Mendoza-Sanchez (6), Equivalence of two models for biodegradation during contaminant transort in groundwater, Water Resour. Res., 4, W46, doi:.9/5wr45. Dentz, M., and B. Berkowitz (3), Transort behavior of a assive solute in continuous time random walks and multirate mass transfer, Water Resour. Res., 39(5),, doi:.9/wr63. Gonzalez-Pinzon, R., R. Haggerty, and D. D. Myrold (), Measuring aerobic resiration in stream ecosystems using the resazurin-resorufin system, J. Geohys. Res., 7, GN6, doi:.9/jg965. Haggerty, R., and S. M. Gorelick (995), Multile-rate mass transfer for modeling diffusion and surface reactions in media with ore-scale heterogeneity, Water Resour. Res., 3(), Haggerty, R., and P. Reeves (), STAMMT-L Version. User s Manual, 76., Sandia Natl. Lab., Albuquerque, N. M. Haggerty, R., S. A. McKenna, and L. C. Meigs (), On the late-time behavior of tracer test breakthrough curves, Water Resour. Res., 36(), Haggerty R., A. Argerich, and E. Martí (8), Develoment of a smart tracer for the assessment of microbiological activity and sediment-water interaction in natural waters: The resazurin-resorufin system, Water Resour. Res., 44, WD, doi:.9/7wr667. Haggerty, R., Martí, E., Argerich, A., von Schiller, D., and Grimm, N. (9), Resazurin as a smart tracer for quantifying metabolically active transient storage in stream ecosystems, J. Geohys. Res., 4, G34, doi:.9/8jg94. Luo, J., O. A. Cirka, M. Dentz, and J. Carrera (8), Temoral moments for transort with mass transfer described by an arbitrary memory function in heterogeneous media, Water Resour. Res., 44(), W5, doi:.9/7wr66. McLaren, A. D. (969), Nitrification in soil: Systems aroaching a steady state, Soil Sci. Soc. Am. J., 33, McMurry, J. E. (), Organic Chemistry: With Biological Alications, 598., Brooks/Cole, Belmont, Calif. Srinivasan, V., and T. P. Clement (8), Analytical solutions for sequentially couled one-dimensional reactive transort roblems. Part I: Mathematical derivations, Adv. Water Resour., 3, 3 8. Sun, Y., and T. A. Buscheck (3), Analytical solutions for reactive transort of N-member radionuclide chains in a single fracture, J. Contam. Hydrol., 6 63, Sun, Y., J. N. Petersen, T. P. Clement, and R. S. Skeen (999), Develoment of analytical solutions for multisecies transort with serial and arallel reactions, Water Resour. Res., 35(), Valocchi, A. J. (985), Validity of the local equilibrium assumtion for modeling sorbing solute transort through homogeneous soils, Water Resour. Res., (6),

slow velocity fast velocity

slow velocity fast velocity Hydrodynamic disersion Disersive transort of contaminants results from the heterogeneous distribution of water flow velocities within and between different soil ores (Figure 1, left). Disersion can be